Question

please explain why the derivative of cos^2(x) = -sin(2x)

Answer 1

\[\frac{d}{dx} cos^2(x) =2cosx (-sinx) = -2sinxcosx \]
By 2sinxcosx = sin(2x), you can get that.

Answer 2

\[\cos^2 (x) \implies [\cos (x)]^2\]
if you apply chain rule you will get \[2[cos(x)]^{2-1} (\frac{d}{dx} \cos x)\]
\[\implies 2[\cos(x)] (-\sin x) \implies -2 \sin x \cos x \implies -\sin (2x)\]